In the field of a high speed digital transmission, a transmission system is known that transmits a block having the CP added to data (a transmitted signal) as shown in FIG. 2. The CP is formed by copying the last K symbols of a data main body part (a useful block time) composed of M symbols before the data main body part.
As such a transmission system, an OFDM (Orthogonal Frequency Division Multiplexing) system or a single carrier block transmission system with cyclic prefix (refer it to as a SC-CP, hereinafter) to which the cyclic prefix is applied is well-known.
Here, a received signal comes under an influence by a channel and a process for removing this influence in a receiver is referred to as an equalizing process. The equalizing process is ordinarily realized by a filter having performance inverse to the transfer function of the channel.
As shown in FIG. 7, in the block transmission system using the cyclic prefix (SC-CP), an equalizer 100 of a discrete frequency domain is used. In this frequency domain equalizer, a received signal vector after the CP is removed is discrete Fourier transformed, multiplied by a weight for each frequency component in a transformed domain and returned again to a signal of a time domain by an inverse discrete Fourier transform to realize an equalization.
The functions of other structures in FIG. 7 will be apparent from the explanation of below-described embodiments.
Non-Patent Document 1 discloses that when a CP is added to a transmitted signal block, an inter-block interference can be removed by frequency domain equalizer (FDE) and a performance of it is improved.
The inter-block interference (refer it also to as an “IBI”, hereinafter) arises in such a way that a delayed signal of a previous block generated in the channel is overlapped on a signal of a current block. When it is assumed that the CP is inserted as the guard interval between the blocks, even if the delayed signal of the previous block remains within the CP of the current block, an influence by the inter-block interference can be removed.
In order to explain the above-described thing in detail below, a process from the transmitting signal to the equalizing process is numerically expressed. Initially, in a transmitter, the transmitted signal is divided into blocks at intervals of M symbols (form a useful block time). In a below-described equation (1), n indicates a number attached to each block.[Equation 1]s(n)=[s0(n),s1(n), . . . , sM-1(n)]T  (1)
The CP is added to the useful block so that a block having the CP is formed and this block is transmitted.[Equation 2]{dot over (s)}(n)=TCPs(n)  (2)
In this case, TCP represents an operation for copying the last K components of the useful block s(n) in the head part in order just as it is.
                    [                  Equation          ⁢                                          ⁢          3                ]                                                                                                                T                CP                            =                              [                                                                                                                              0                                                      K                            ×                                                          (                                                              M                                -                                K                                                            )                                                                                                      ⁢                                                  I                          K                                                                                                                                                                        I                        M                                                                                            ]                                                                        Matrix              ⁢                                                          ⁢              size              ⁢                              :                            ⁢                                                          ⁢                              (                                  M                  +                  K                                )                            ×              M                                                          (        3        )            
Assuming that an impulse response of the channel (a communication path) is h={h0, h1, . . . , hL}, a received signal block in the receiver is expressed by a below-described equation.
                    [                  Equation          ⁢                                          ⁢          4                ]                                                                                                                                                                     r                    .                                    ⁡                                      (                    n                    )                                                  =                                ⁢                                                      [                                                                                                                        r                            .                                                    0                                                ⁡                                                  (                          n                          )                                                                    ,                      …                      ⁢                                                                                          ,                                                                                                    r                            .                                                                                M                            +                            K                            -                            1                                                                          ⁡                                                  (                          n                          )                                                                                      ]                                    T                                                                                                        =                            ⁢                              H                ⁡                                  [                                                                                                                                          s                            .                                                    ⁡                                                      (                                                          n                              -                              1                                                        )                                                                                                                                                                                                                    s                            .                                                    ⁡                                                      (                            n                            )                                                                                                                                ]                                                                                        (        4        )            
In this case, H is expressed as shown below.
                    [                  Equation          ⁢                                          ⁢          5                ]                                                                      H          =                      [                                                            0                                                  …                                                                      h                    L                                                                    …                                                                      h                    0                                                                    0                                                  …                                                  0                                                                              ⋮                                                                                                                                          ⋱                                                  ⋱                                                                                                                                          ⋱                                                  ⋱                                                  ⋮                                                                              ⋮                                                                                                                                                                                                                                  ⋱                                                  ⋱                                                                                                                                          ⋱                                                  0                                                                              0                                                  …                                                  …                                                  …                                                  0                                                                      h                    L                                                                    …                                                                      h                    0                                                                        ]                          ⁢                                  ⁢                  Matrix          ⁢                                          ⁢          size          ⁢                      :                    ⁢                                          ⁢                      (                          M              +              K                        )                    ×          2          ⁢                      (                          M              +              K                        )                                              (        5        )            
Further, H is decomposed to two submatrices of (M+K)×(M+K),
                    [                  Equation          ⁢                                          ⁢          6                ]                                                                      H          1                =                  [                                                    0                                            …                                                              h                  L                                                            …                                                              h                  1                                                                                    ⋮                                                                                                                          ⋱                                            ⋱                                            ⋮                                                                    ⋮                                                                                                                                                                                                        ⋱                                                              h                  L                                                                                    ⋮                                                                                                                                                                                                                                                                                      ⋮                                                                    0                                            …                                            …                                            …                                            0                                              ]                                    (        6        )                                [                  Equation          ⁢                                          ⁢          7                ]                                                                      H          0                =                  [                                                                      h                  0                                                                                                                                                                                                                                                                                                                                                                                                            ⋮                                                              h                  0                                                                                                                                          0                                                                                                                                                                    h                  L                                                                                                                                          ⋱                                                                                                                                                                                                                                                                                                              ⋱                                                                                                                          ⋱                                                                                                                                                  0                                                                                                                                            h                  L                                                            …                                                              h                  0                                                              ]                                    (        7        )            the received signal block is expressed as described below.
                    [                  Equation          ⁢                                          ⁢          8                ]                                                                                                                                  r                  .                                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    H                    1                                    ⁢                                                            s                      .                                        ⁡                                          (                                              n                        -                        1                                            )                                                                      +                                                      H                    0                                    ⁢                                                            s                      .                                        ⁡                                          (                      n                      )                                                                      +                                                      n                    .                                    ⁡                                      (                    n                    )                                                                                                                          =                            ⁢                                                                    H                    1                                    ⁢                                      T                    CP                                    ⁢                                      s                    ⁡                                          (                                              n                        -                        1                                            )                                                                      +                                                      H                    0                                    ⁢                                      T                    CP                                    ⁢                                      s                    ⁡                                          (                      n                      )                                                                      +                                                      n                    .                                    ⁡                                      (                    n                    )                                                                                                          (        8        )                                Matrix        ⁢                                  ⁢        size        ⁢                  :                ⁢                  (                      M            +            K                    )                ×        1                                        Here, a first term of the right side of the equation (8) shows a signal component from a (n−1) th transmitting block and represents a component of the inter-block interference (IBI).
At a receiver, the CP is removed. This is expressed by a below-described equation.
                    [                  Equation          ⁢                                          ⁢          9                ]                                                                                                                r                ⁡                                  (                  n                  )                                            =                            ⁢                                                R                  CP                                ⁢                                                      r                    .                                    ⁡                                      (                    n                    )                                                                                                                          =                            ⁢                                                                    R                    CP                                    ⁢                                      H                    1                                    ⁢                                      T                    CP                                    ⁢                                      s                    ⁡                                          (                                              n                        -                        1                                            )                                                                      +                                                      R                    CP                                    ⁢                                      H                    0                                    ⁢                                      T                    CP                                    ⁢                                      s                    ⁡                                          (                      n                      )                                                                      +                                                      R                    CP                                    ⁢                                                            n                      .                                        ⁡                                          (                      n                      )                                                                                                                              (        9        )                                Matrix        ⁢                                  ⁢        size        ⁢                  :                ⁢        M        ×        1                                        
In this case,[Equation 10]RCP=[0M×K IM] Matrix size: M×(M+K)  (10)
At this time, for the length K of the CP and the order number L of the channel (physically corresponding to the impulse response length of the channel), K≧L is assumed. That is, when the length K of the CP is equal to or larger than the order number L of the channel, since RCP H1=0, irrespective of the transmitted signal block, the received signal r(n) after the CP is removed is expressed by a below-described equation and the inter-block interference component is removed.
                    [                  Equation          ⁢                                          ⁢          11                ]                                                                                                                r                ⁡                                  (                  n                  )                                            =                            ⁢                                                                    R                    CP                                    ⁢                                      H                    0                                    ⁢                                                            s                      .                                        ⁡                                          (                      n                      )                                                                      +                                                      R                    CP                                    ⁢                                                            n                      .                                        ⁡                                          (                      n                      )                                                                                                                                              =                            ⁢                                                                    R                    CP                                    ⁢                                      H                    0                                    ⁢                                      T                    CP                                    ⁢                                      s                    ⁡                                          (                      n                      )                                                                      +                                                      R                    CP                                    ⁢                                                            n                      .                                        ⁡                                          (                      n                      )                                                                                                                              (        11        )            
When RCPH0TCP of the equation (11) is expanded, a below-described equation is obtained.
                    [                  Equation          ⁢                                          ⁢          12                ]                                                                                  R            CP                    ⁢                      H            0                    ⁢                      T            CP                          =                              [                                                                                h                    0                                                                    0                                                  …                                                  0                                                                      h                    L                                                                    …                                                                      h                    1                                                                                                ⋮                                                                      h                    0                                                                    ⋱                                                                                                                                          ⋱                                                  ⋱                                                  ⋮                                                                                                  h                    L                                                                                                                                                            ⋱                                                  ⋱                                                                                                                                          ⋱                                                                      h                    L                                                                                                0                                                  ⋱                                                                                                                                          ⋱                                                  ⋱                                                                                                                                          0                                                                              ⋮                                                  ⋱                                                  ⋱                                                                                                                                          ⋱                                                  ⋱                                                  ⋮                                                                              ⋮                                                                                                                                          ⋱                                                  ⋱                                                                                                                                          ⋱                                                  0                                                                              0                                                  …                                                  …                                                  0                                                                      h                    L                                                                    …                                                                      h                    0                                                                        ]                    =                      C            CP                                              (        12        )            
A matrix having such a structure as shown in the equation (12) is called a circulant matrix which can be transformed using a unitary similarity by a “discrete Fourier transform” (DFT) matrix.
When the nature of the circulant matrix is used, a below-described equation can be obtained.[Equation 13]CCP=DHΛD  (13)
In this case,
                    [                  Equation          ⁢                                          ⁢          14                ]                                                            Λ        =                              [                                                                                λ                    0                                                                                                ⋮                                                                                                  λ                                          M                      -                      1                                                                                            ]                    =                      D            ⁡                          [                                                                                          h                      0                                                                                                            ⋮                                                                                                              h                      L                                                                                                                                  0                                              (                                                  M                          -                          L                          -                          1                                                )                                                                                                        ]                                                          (        14        )                                [                  Equation          ⁢                                          ⁢          15                ]                                                            D        =                  [                                                    1                                            1                                            …                                            1                                                                    1                                                              ⅇ                                                            -                      j                                        ⁢                                                                  2                        ⁢                        π                        ×                        1                        ×                        1                                            M                                                                                                  …                                                              ⅇ                                                            -                      j                                        ⁢                                                                  2                        ⁢                        π                        ×                        1                        ×                                                  (                                                      M                            -                            1                                                    )                                                                    M                                                                                                                          ⋮                                            ⋮                                            ⋱                                            ⋮                                                                    1                                                              ⅇ                                                            -                      j                                        ⁢                                                                  2                        ⁢                        π                        ×                                                  (                                                      M                            -                            1                                                    )                                                ×                        1                                            M                                                                                                  …                                                              ⅇ                                                            -                      j                                        ⁢                                                                  2                        ⁢                        π                        ×                                                  (                                                      M                            -                            1                                                    )                                                ×                                                  (                                                      M                            -                                                    )                                                ⁢                        1                                            M                                                                                                    ]                                    (        15        )            
When a noise component of a second term of the right side of the equation (11) is replaced by n(n), the received signal r(n) after the CP is removed can be expressed as described below.[Equation 16]r(n)=DHΛDs(n)+n(n)  (16)
The equalizing process after the CP is removed is expressed by an equation as described below. The frequency domain equalizer uses discrete Fourier transform for the received signal block after the CP is removed, multiplies the received signal block by a weight for each frequency component in the frequency domain and returns again to a signal of a time domain by an inverse discrete Fourier transform. Assuming that the discrete frequency domain weight is expressed as a diagonal matrix, whose diagonal element is {γ0, . . . , γM-1}, an output of the equalizer is expressed by a below-described equation.
                    [                  Equation          ⁢                                          ⁢          17                ]                                                                                                                                  s                  ^                                ⁡                                  (                  n                  )                                            =                            ⁢                                                (                                                            D                      H                                        ⁢                    Γ                    ⁢                                                                                  ⁢                    D                                    )                                ⁢                                  D                  H                                ⁢                                  ΛDs                  ⁡                                      (                    n                    )                                                                                                                          =                            ⁢                                                D                  H                                ⁢                Γ                ⁢                                                                  ⁢                Λ                ⁢                                                                  ⁢                                  Ds                  ⁡                                      (                    n                    )                                                                                                          (        17        )            
Further, Non-Patent Document 1 discloses an equalizer weight of a zero forcing (ZF) criterion and an equalizer weight of a minimum mean-square-error (MMSE) criterion.
ZF Equalizer Weight
                    [                  Equation          ⁢                                          ⁢          18                ]                                                                                  γ            i                    =                      1                          λ              i                                      ,                                  ⁢                  i          =          0                ,        …        ⁢                                  ,                  M          -          1                                    (        18        )            MMSE Equalizer Weight
                    [                  Equation          ⁢                                          ⁢          19                ]                                                                                  γ            i                    =                                    λ              i              *                                                                                                              λ                    i                                                                    2                            +                                                σ                  n                  2                                /                                  σ                  s                  2                                                                    ,                  i          =          0                ,        …        ⁢                                  ,                  M          -          1                                    (        19        )            σs2 is a variance of a signal s(n) and, σn2 is a variance of noise n(n)
Here, Λ denotes the diagonal matrix with diagonal element which is {λ0, . . . , λM-1} of the discrete Fourier transform of the impulse response of the channel obtained from the equation (14). Simulation examples using these weights are also disclosed in the Non-Patent Document 1 (FIG. 8). The MMSE criterion equalizer of the single carrier block transmission system is more excellent in its performance than the ZF criterion equalizer.
A main reason why the ZF criterion equalizer is inferior to the MMSE criterion equalizer is a noise enhancement. The noise enhancement is a phenomenon that when a channel response of a communication path in a certain frequency i (0≦i≦M−1) is 0 or near to 0, a weight in the frequency i shows a very large value so that noise is amplified.    Non-Patent Document 1: Kazunori Hayashi “Fundamentals of Modulation/Demodulation and Equalization Technologies) Proc. MWE2004, pp. 523-532, 2004.